Math

Digital Tool Dependency vs. Conceptual Understanding in Secondary Mathematics Presenters 

Nayeli Cordova
Our Lady of the Lake University

This qualitative study explores the intersection of digital tool proficiency and conceptual mathematical understanding in secondary education. As digital graphing tools like Desmos become integral to state-tested courses, there is a growing concern regarding student dependency on these tools at the expense of foundational reasoning. The purpose of this research is to investigate how mathematics teachers perceive this shift and its impact on student mastery. The primary goal is to answer how educators perceive the impact of digital graphing tools on conceptual understanding, the extent of tool dependency observed during high-stakes testing preparation, and what instructional strategies are used to ensure digital proficiency does not replace mathematical reasoning. Data will be collected through semi-structured interviews with 5-8 secondary mathematics educators recruited via purposeful sampling within the San Antonio area. Interviews will be conducted either in person or virtually via Microsoft Teams to ensure secure data management. To ensure minimal risk, all data will be de-identified using pseudonyms. The findings aim to provide insight into balancing instructional technology with traditional mathematical rigor, helping educators design strategies that leverage technology without compromising foundational problem-solving skills. 

New Construction of DG Modules Pertaining to a Conjecture by Carlsson on Singular Homology of Certain CW Complexes

Austin Brown
University of Texas at Arlington

This project expands a method by Iyengar and Walker for constructing DG modules over certain fields (potentially with characteristic greater than 0) whose homology has length less than a certain power of two. In particular, let R be a regular local ring that is a complete intersection with residue field k. There exists an induced map on DG K-modules – K the Koszul complex on a minimal generating set of the maximal ideal of R – whose mapping cone has finite length. This mapping cone induces a long exact sequence on homology which then induces an exact sequence of graded R-modules whose middle term has length less than 2r, where r is the Krull dimension of R. The new examples presented here are interesting in context of a conjecture by Gunnar Carlsson in the ‘80s. This conjecture states that if X is a finite CW complex, E is an elementary abelian group, and X admits a free, cellular E-action, then the total rank of its singular homology with Z/(p)-coefficients is at least 2r. It is currently unknown if such a CW complex exists, but if any example constructed here arises as the singular homology of a CW complex with a free, cellular E-action on X, then Carlsson’s conjecture is resolved.